A299918 Motzkin numbers (A001006) mod 8.

1, 1, 2, 4, 1, 5, 3, 7, 3, 3, 4, 6, 7, 3, 2, 4, 3, 3, 6, 4, 3, 7, 7, 7, 5, 5, 4, 2, 1, 5, 3, 7, 3, 3, 6, 4, 3, 7, 1, 5, 1, 1, 4, 2, 5, 1, 4, 6, 5, 5, 2, 4, 5, 1, 1, 1, 3, 3, 4, 6, 7, 3, 2, 4, 3, 3, 6, 4, 3, 7, 1, 5, 1, 1, 4, 2, 5, 1, 6, 4, 1, 1, 2, 4, 1

Offset: 0

Author

N. J. A. Sloane, Mar 16 2018

Keywords

• nonn
• hear

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• S.-P. Eu, S.-C. Liu, and Y.-N. Yeh, Catalan and Motzkin numbers modulo 4 and 8, Europ. J. Combin. 29 (2008), 1449-1466.
• Christian Krattenthaler and Thomas W. Müller, Motzkin numbers and related sequences modulo powers of 2, arXiv:1608.05657 [math.CO], 2016-2018.
• E. Rowland and R. Yassawi, Automatic congruences for diagonals of rational functions, J. Théorie Nombres Bordeaux 27 (2015), 245-288.
• Ying Wang and Guoce Xin, A Classification of Motzkin Numbers Modulo 8, Electron. J. Combin., 25(1) (2018), #P1.54.

Crossrefs

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

Programs

Maple

f:= rectoproc({(3+3*n)*a(n)+(5+2*n)*a(1+n)+(-4-n)*a(n+2), a(0) = 1, a(1) = 1}, a(n), remember): seq(f(n) mod 8, n=0..200); # Robert Israel, Mar 16 2018

Mathematica

Table[Mod[GegenbauerC[n, -n - 1, -1/2] / (n + 1), 8], {n, 0, 100}] (* Vincenzo Librandi, Sep 08 2018 *)

PARI

catalan(n) = binomial(2*n, n)/(n+1);
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*catalan(k+1)) % 8; \\ Michel Marcus, May 23 2022